Question: All of the 4th grade teachers and students from Santa Rita went on a field trip to an archaeology museum. Tickets were $$6.50$ each for teachers and $$4.00$ each for students, and the group paid $$39.50$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$19.50$ each for teachers and $$12.50$ each for students, and the group paid $$121.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6.5x+4y = 39.5}$ ${19.5x+12.5y = 121}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-19.5x-12y = -118.5}$ ${19.5x+12.5y = 121}$ Add the top and bottom equations together. $ 0.5y = 2.5 $ $ y = \dfrac{2.5}{0.5}$ ${y = 5}$ Now that you know ${y = 5}$ , plug it back into $ {6.5x+4y = 39.5}$ to find $x$ ${6.5x + 4}{(5)}{= 39.5}$ $6.5x+20 = 39.5$ $6.5x = 19.5$ $x = \dfrac{19.5}{6.5}$ ${x = 3}$ You can also plug ${y = 5}$ into $ {19.5x+12.5y = 121}$ and get the same answer for $x$ ${19.5x + 12.5}{(5)}{= 121}$ ${x = 3}$ There were $3$ teachers and $5$ students on the field trips.